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Sum of inverse squares of natural numbers
The Basel problem is a problem in mathematical analysis with relevance to number theory, concerning an infinite sum of inverse squares. It was first posed
Basel_problem
City in Switzerland
the so-called 'Basel Problem' was solved in the city, which is regarded as one of the most important achievements in mathematics. Basel was ranked the
Basel
Number, approximately 3.14
simple solution for this infinite series was a famous problem in mathematics called the Basel problem. Leonhard Euler solved it in 1735 when he showed it
Pi
Negative integer two units from the origin in mathematics
positive; the inverse-square law; grid turbulence decay; and the Basel problem. The Basel problem states that the sum of the square reciprocals of natural numbers
−2
Summation formula
}}+\int _{m}^{n}f^{(7)}(x){\frac {P_{7}(x)}{7!}}\,dx.\end{aligned}}} The Basel problem is to determine the sum 1 + 1 4 + 1 9 + 1 16 + 1 25 + ⋯ = ∑ n = 1 ∞
Euler–Maclaurin_formula
Analytic function in mathematics
computed by Euler. The first of them, ζ(2), provides a solution to the Basel problem. In 1979, Roger Apéry proved the irrationality of ζ(3), and as a consequence
Riemann_zeta_function
Problem in probability theory
{1}{1^{2}}}+{\frac {1}{2^{2}}}+\cdots +{\frac {1}{n^{2}}}+\cdots } (see Basel problem). Bound the desired probability using the Chebyshev inequality: P
Coupon_collector's_problem
Number divisible only by 1 and itself
Leonhard Euler and his first major result, the solution to the Basel problem. The problem asked for the value of the infinite sum 1 + 1 4 + 1 9 + 1 16 +
Prime_number
Swiss mathematician (1707–1783)
accomplishments, solving several unsolved problems in number theory and analysis, including the famous Basel problem. Euler has also been credited for discovering
Leonhard_Euler
"Euler's number"). The sum of the reciprocals of the square numbers (the Basel problem) is the transcendental number π2 /6 , or ζ(2) where ζ is the Riemann
List_of_sums_of_reciprocals
Uses of the constant
{1}{3^{2}}}+{\frac {1}{4^{2}}}+\cdots ={\frac {\pi ^{2}}{6}}} (see also Basel problem and Riemann zeta function) ζ ( 4 ) = 1 1 4 + 1 2 4 + 1 3 4 + 1 4 4 +
List_of_formulae_involving_π
1897 proposed law to define squaring the circle
is not considered a solution to the ancient problem of squaring the circle. This is because the problem is to construct the area using a compass and
Indiana_pi_bill
Conjecture on zeros of the zeta function
values of s {\displaystyle s} , in conjunction with his solution to the Basel problem. He also proved that it equals the Euler product ζ ( s ) = ∏ p prime
Riemann_hypothesis
Perimeter of a circle or ellipse
A History of Pi In culture Indiana pi bill Pi Day Related topics Squaring the circle Basel problem Six nines in π Other topics related to π Tau v t e
Circumference
Infinite series with alternating signs
n = 0 respectively. This line of research extended his work on the Basel problem and leading towards the functional equations of what are now known as
1_−_2_+_3_−_4_+_⋯
Array of line segments normal to points of a square lattice
{x}{x+y}},{\frac {y}{x+y}},{\frac {1}{x+y}}\right).} The solution to the Basel problem can be used to show that the proportion of points in the n × n {\displaystyle
Euclid's_orchard
Value approached by a mathematical object
Otherwise, the series is said to be divergent. A classic example is the Basel problem, where a n = 1 / n 2 {\displaystyle a_{n}=1/n^{2}} . Then ∑ n = 1 ∞
Limit_(mathematics)
Banking regulation framework
Basel III is the third of three Basel Accords, a framework that sets international standards and minimums for bank capital requirements, stress tests
Basel_III
Decomposition of periodic functions
method of solving the heat problem was made possible by Fourier's work. Another application is to solve the Basel problem by using Parseval's theorem
Fourier_series
.. 1/6 One sixth. Often appears in mathematical equations, such as in the sum of squares of the integers and in the solution to the Basel problem.
List_of_numbers
Welsh mathematician (1675–1749)
A History of Pi In culture Indiana pi bill Pi Day Related topics Squaring the circle Basel problem Six nines in π Other topics related to π Tau v t e
William_Jones_(mathematician)
Index of articles associated with the same name
the squares of the first three equals the square of the fourth. The Basel problem, solved by Euler in terms of π {\displaystyle \pi } , asked for an exact
Sum_of_squares
Concept of complex analysis
}{\frac {1}{n^{2}}}={\frac {\pi ^{2}}{6}}} which is a proof of the Basel problem. The same argument works for all f ( x ) = x − 2 n {\displaystyle f(x)=x^{-2n}}
Residue_theorem
{(-1)^{n}z^{2n+1}}{2n+1}}.} His use of power series enabled him to solve the famous Basel problem in 1735: lim n → ∞ ( 1 1 2 + 1 2 2 + 1 3 2 + ⋯ + 1 n 2 ) = π 2 6 . {\displaystyle
Contributions of Leonhard Euler to mathematics
Contributions_of_Leonhard_Euler_to_mathematics
Annual mathematical celebration on March 14
A History of Pi In culture Indiana pi bill Pi Day Related topics Squaring the circle Basel problem Six nines in π Other topics related to π Tau v t e
Pi_Day
Topics referred to by the same term
banking capital Basel Convention, a treaty to reduce the movements of hazardous waste between nations Basel problem, a famous problem in number theory
Basel_(disambiguation)
Two numbers without shared prime factors
example of an Euler product, and the evaluation of ζ(2) as π2/6 is the Basel problem, solved by Leonhard Euler in 1735. There is no way to choose a positive
Coprime_integers
Approximations of π Arithmetic–geometric mean Bailey–Borwein–Plouffe formula Basel problem Borwein's algorithm Buffon's needle Cadaeic Cadenza Chronology of computation
List_of_topics_related_to_π
Signed odd unit fractions sum to π/4
A History of Pi In culture Indiana pi bill Pi Day Related topics Squaring the circle Basel problem Six nines in π Other topics related to π Tau v t e
Leibniz_formula_for_π
Problem of constructing equal-area shapes
Squaring the circle is a problem in geometry first proposed in Greek mathematics. It is the challenge of constructing a square with the area of a given
Squaring_the_circle
French mathematician (1916–1994)
{1}{27}}+{\frac {1}{64}}+\cdots \neq {\frac {p}{q}}} Apéry's constant Basel problem Apéry, François (1996). "Roger Apéry, 1916-1994: A Radical Mathematician"
Roger_Apéry
Sum of the inverses of the positive cubes
of computers and to algorithmic improvements. Riemann zeta function Basel problem — ζ(2) Catalan's constant List of sums of reciprocals Wedeniwski (2001)
Apéry's_constant
Fixed number that has received a name
and elementary functions originated when Euler famously solved the Basel problem by giving ζ ( 2 ) = 1 6 π 2 {\displaystyle \zeta (2)={\frac {1}{6}}\pi
Mathematical_constant
Infinite products of functions indexed by primes
\zeta (2)={\frac {\pi ^{2}}{6}}} by the previous product, known as Basel problem, one finds that π 2 = ( ∏ p ≡ 1 ( mod 4 ) p p + 1 ) ( ∏ p ≡ 3 ( mod
Euler_product
Type of figurate number
polygonal number Polyhedral number Fermat polygonal number theorem See Basel problem. Tattersall, James J. (2005). Elementary Number Theory in Nine Chapters
Polygonal_number
Difference between logarithm and harmonic series
series of inverse triangular numbers also features in the study of the Basel problem posed by Pietro Mengoli. Mengoli proved that ∑ k = 1 ∞ 1 2 T k = 1 {\displaystyle
Euler's_constant
Concept in geometry
pieces into a square of equal area. This is called Tarski's circle-squaring problem. The nature of Laczkovich's proof is such that it proves the existence
Area_of_a_circle
Mathematical power series of arctangent
Madhava–Gregory–Leibniz series: Benko, David; Molokach, John (2013). "The Basel Problem as a Rearrangement of Series". College Mathematics Journal. 44 (3):
Arctangent_series
Special mathematical function defined as sin(x)/x
this series to the expansion of the infinite product form to solve the Basel problem. The product of 1-D sinc functions readily provides a multivariate sinc
Sinc_function
Mathematical series with a finite sum
The reciprocals of square numbers produce a convergent series (the Basel problem): 1 1 + 1 4 + 1 9 + 1 16 + 1 25 + 1 36 + ⋯ = π 2 6 . {\displaystyle
Convergent_series
1998 mathematics book by Aigner and Ziegler
\pi } (pi) is irrational. Chapter 9: Four proofs of the solution to Basel problem, namely that ∑ n ≥ 1 ∞ 1 n 2 = π 2 6 {\displaystyle \sum _{n\geq 1}^{\infty
Proofs_from_THE_BOOK
Divergent sum of positive unit fractions
π 2 / 6 {\displaystyle \zeta (2)=\pi ^{2}/6} , the solution to the Basel problem, Apéry's constant ζ ( 3 ) {\displaystyle \zeta (3)} , proved by Roger
Harmonic_series_(mathematics)
Hobson, Ernest William (1913). 'Squaring the Circle': a History of the Problem (PDF). Cambridge University Press. p. 27. Yoshio, Mikami; Eugene Smith
Chronology of computation of pi
Chronology_of_computation_of_pi
π 2 / 6 {\displaystyle \zeta (2)=\pi ^{2}/6} is the solution to the Basel problem, the Airy zeta function may be exactly evaluated at s = 2: ζ A i ( 2
Airy_zeta_function
Sum of the first n whole number reciprocals; 1/1 + 1/2 + 1/3 + ... + 1/n
ISBN 978-0-201-89683-1. Ed Sandifer, How Euler Did It — Estimating the Basel problem Archived 2005-05-13 at the Wayback Machine (2003) Paule, Peter; Schneider
Harmonic_number
Conditions for switching order of integration in calculus
g(x)\,g(xy)\,\mathrm {d} x\,\mathrm {d} y} This can be applied to the Basel problem: π 2 4 = ( arcsin ( 1 ) ) 2 = [ ∫ 0 1 1 1 − x 2 d x ] 2 = ∫ 0 1 ∫
Fubini's_theorem
Book by Hiroshi Yūki
independently come up with an idea that leads to an elegant solution to the Basel problem, they become close friends. She has a deep knowledge of mathematics
Math_Girls
the primes diverges Banach fixed-point theorem Banach–Tarski paradox Basel problem Bolzano–Weierstrass theorem Brouwer fixed-point theorem Buckingham π
List_of_mathematical_proofs
German mathematician (1690–1764)
solutions to Goldbach. Also, in 1729 Goldbach closely approximated the Basel problem, which prompted Euler's interest and concurring breakthrough solution
Christian_Goldbach
Misunderstanding in Japanese education
until you are satisfied,' but if teachers are like this, it must be a problem before educational reform." Other issues with treating pi as approximately
Pi_is_3
Mathematical criterion about whether a series converges
diverges. The case of p = 2 , k = 1 {\displaystyle p=2,k=1} is the Basel problem and the series converges to π 2 6 {\displaystyle {\frac {\pi ^{2}}{6}}}
Convergence_tests
Function that quantifies how near a number is to being rational
π". In Chudnovsky, David V.; Chudnovsky, Gregory V. (eds.). The Riemann Problem, Complete Integrability and Arithmetic Applications. Lecture Notes in Mathematics
Irrationality_measure
Constants of the mathematical zeta function
The computation of ζ ( 2 ) {\displaystyle \zeta (2)} is known as the Basel problem. The value of ζ ( 4 ) {\displaystyle \zeta (4)} is related to the Stefan–Boltzmann
Particular values of the Riemann zeta function
Particular_values_of_the_Riemann_zeta_function
Cartwright, but that she had not traced its origin. It still remains on the 4th problem sheet today for the Analysis IA course at Cambridge University. Consider
Proof_that_pi_is_irrational
theory) Barban–Davenport–Halberstam theorem (analytic number theory) Basel problem (mathematical analysis) Beatty's theorem (Diophantine approximation)
List_of_theorems
(2)=\sum _{k=1}^{\infty }{\frac {1}{k^{2}}}={\frac {\pi ^{2}}{6}}} (the Basel problem) ζ ( 4 ) = ∑ k = 1 ∞ 1 k 4 = π 4 90 {\displaystyle \zeta (4)=\sum _{k=1}^{\infty
List_of_mathematical_series
1897 event in Basel, Switzerland
congress of the Zionist Organization (ZO) held in the Stadtcasino Basel in the city of Basel on August 29–31, 1897. Two hundred and eight delegates from 17
First_Zionist_Congress
functions Apéry's constant - the solution to ζ(3) Artin conjecture Basel problem boils down to ζ(2) Birch and Swinnerton-Dyer conjecture Riemann hypothesis
List_of_zeta_functions
Japanese mathematician (c. 1642–1708)
in Japan. He successfully applied it to problems suggested by his contemporaries. Before him, these problems were solved using arithmetical methods. In
Seki_Takakazu
} Therefore, 22/7 > π. The evaluation of this integral was the first problem in the 1968 Putnam Competition. That the integral is positive follows from
Proof_that_22/7_exceeds_π
Ghanaian footballer (born 2003)
during the winter break, he was brought up to Basel's first team for the second half of their 2022–23 FC Basel season. However, apart from test matches, he
Jonas_Adjetey
Texts belonging to the Śrauta ritual
similar interest and approach to doubling and other geometric transformation problems. Abraham Seidenberg, followed by Bartel Leendert van der Waerden, sees
Shulba_Sutras
Chinese mathematician-astronomer (429–500)
A History of Pi In culture Indiana pi bill Pi Day Related topics Squaring the circle Basel problem Six nines in π Other topics related to π Tau v t e
Zu_Chongzhi
Proofs of Fermat's theorem on sums of two squares Riemann zeta function Basel problem on ζ(2) Hurwitz zeta function Bernoulli number Agoh–Giuga conjecture
List_of_number_theory_topics
One over a whole number
{1}{5}}-{\frac {1}{7}}+{\frac {1}{9}}-\cdots ={\frac {\pi }{4}}.} The Basel problem concerns the sum of the square unit fractions: 1 + 1 4 + 1 9 + 1 16
Unit_fraction
Group of Vedic Sanskrit texts
square produces an area double the size of the original square. Another problem tackled by Baudhāyana is that of finding a circle whose area is the same
Baudhayana_sutras
Varying methods used to calculate pi
value of Pi". Rather, the bill dealt with a purported solution to the problem of geometrically "squaring the circle". The bill was nearly passed by the
Approximations_of_pi
Swiss 14th-century pogrom
The Basel Massacre was an anti-Semitic massacre in Basel, which occurred in 1349 in connection with alleged well poisoning as part of the Black Death persecutions
Basel_Massacre
1970 book by Petr Beckmann
A History of Pi In culture Indiana pi bill Pi Day Related topics Squaring the circle Basel problem Six nines in π Other topics related to π Tau v t e
A_History_of_Pi
Mnemonics of pi's digits
who could estimate its value? For me, your problem had equal advantages. Long ago, mysterious, a problem blocked All the admirable process, the great
Piphilology
English mathematician (1686–1751)
A History of Pi In culture Indiana pi bill Pi Day Related topics Squaring the circle Basel problem Six nines in π Other topics related to π Tau v t e
John_Machin
Theorem in transcendental number theory
function j was conjectured by Daniel Bertrand in 1997, and remains an open problem. Writing q = e2πiτ for the square of the nome and j(τ) = J(q), the conjecture
Lindemann–Weierstrass_theorem
Swiss footballer (born 1995)
previously played for the club's second team. He moved to first tier FC Basel in 2015, where he won consecutive league titles in 2015–16 and 2016–17,
Manuel_Akanji
Infinite sum
of this series is 1 6 π 2 {\textstyle {\frac {1}{6}}\pi ^{2}} ; see Basel problem. This type of bounding strategy is the basis for general series comparison
Series_(mathematics)
Calendar year
Leonhard Euler solves the Basel problem, first posed by Pietro Mengoli in 1644, and the Seven Bridges of Königsberg problem. The King's Highway (Charleston
1735
Banking regulation framework
Basel II is the second of the Basel Accords, which are recommendations on banking laws and regulations issued by the Basel Committee on Banking Supervision
Basel_II
Type of figurate number constructed by combining heptagons
x={\frac {5n^{2}-3n}{2}}} for its unique positive root n. "Beyond the Basel Problem: Sums of Reciprocals of Figurate Numbers" (PDF). Archived from the original
Heptagonal_number
Ratio of the perimeter of Bernoulli's lemniscate to its diameter
{3}}}}{\dfrac {\varpi }{12^{1/8}\pi }}.} In a spirit similar to that of the Basel problem, ∑ z ∈ Z [ i ] ∖ { 0 } 1 z 4 = G 4 ( i ) = ϖ 4 15 {\displaystyle \sum
Lemniscate_constant
Set of integers whose sum of reciprocals diverges
2, 3, ...) is a large set. The set of square numbers is small (see Basel problem § The Riemann zeta function). So is the set of cube numbers, the set
Large_set_(combinatorics)
Italian mathematician (1626–1686)
Society. Mengoli died in Bologna in 1685. Mengoli first posed the famous Basel problem in 1650, solved in 1735 by Leonhard Euler. In 1650, he also proved that
Pietro_Mengoli
Ecumenical Council of the Catholic Church (1431–1449)
of the Lateran in Rome's Lateran Palace). It was convoked in Basel as the Council of Basel by Pope Martin V shortly before his death in February 1431 and
Council_of_Florence
A History of Pi In culture Indiana pi bill Pi Day Related topics Squaring the circle Basel problem Six nines in π Other topics related to π Tau v t e
Madhava's_correction_term
Mathematician
A History of Pi In culture Indiana pi bill Pi Day Related topics Squaring the circle Basel problem Six nines in π Other topics related to π Tau v t e
John_Wrench
Japanese mathematician (1949–2020)
A History of Pi In culture Indiana pi bill Pi Day Related topics Squaring the circle Basel problem Six nines in π Other topics related to π Tau v t e
Yasumasa_Kanada
Swiss footballer (born 1988)
and later of Concordia Basel. He moved to the Basel youth setup in 2003. He signed his first professional contract with Basel in 2005. He became the first-choice
Yann_Sommer
Russo–Vallois integral Stratonovich integral Skorokhod integral Miscellaneous Basel problem Euler–Maclaurin formula Gabriel's horn Integration Bee Proof that 22/7
Common integrals in quantum field theory
Common_integrals_in_quantum_field_theory
Fastest curve descent without friction
Springer Basel Aktiengesellschaft. pp. 117–118. ISBN 978-3-0348-5068-1. Babb, Jeff; Currie, James (July 2008), "The Brachistochrone Problem: Mathematics
Brachistochrone_curve
following is a list of notable unsolved problems grouped into broad areas of physics. Some of the major unsolved problems in physics are theoretical, meaning
List of unsolved problems in physics
List_of_unsolved_problems_in_physics
Theorem in number theory
_{k=1}^{\infty }{\frac {1}{k^{2}}}={\frac {\pi ^{2}}{6}}} (see the Basel problem), the above constant log 5/3 = 0.51082... can be improved to log π2/6
Divergence of the sum of the reciprocals of the primes
Divergence_of_the_sum_of_the_reciprocals_of_the_primes
1735—Leonhard Euler solves the Basel problem, relating an infinite series to π. 1736—Leonhard Euler solves the problem of the Seven Bridges of Königsberg
Timeline_of_mathematics
Criterion for the convergence of a series
series (1 + 1 + 1 + 1 + ⋯) diverges, the second (the one central to the Basel problem) converges absolutely and the third (the alternating harmonic series)
Ratio_test
subsequence with respect to the sup norm. Bargmann Bargmann transform Basel The Basel problem says ∑ 1 ∞ 1 n 2 = π 2 6 . {\displaystyle \sum _{1}^{\infty }{\frac
Glossary of real and complex analysis
Glossary_of_real_and_complex_analysis
Mathematical series in trigonometry
Madhava–Gregory–Leibniz series: Benko, David; Molokach, John (2013). "The Basel Problem as a Rearrangement of Series". College Mathematics Journal. 44 (3):
Madhava_series
Canadian soccer player (born 1999)
senior appearance for Liverpool in the FA Cup in 2020. In 2021, he joined Basel of the Swiss Super League for a reported fee of £1.3 million. After a loan
Liam_Millar
before observing it Solution to the Basel problem (1735) Jacob Bernoulli's work: Ars Conjectandi published in Basel in 1713, theory of probability from
List of Swiss inventions and discoveries
List_of_Swiss_inventions_and_discoveries
Leonhard Euler solves the Basel problem, first posed by Pietro Mengoli in 1644, and the Seven Bridges of Königsberg problem. May 22 – George Hadley publishes
1735_in_science
3rd century calculation of π by Liu Hui
A History of Pi In culture Indiana pi bill Pi Day Related topics Squaring the circle Basel problem Six nines in π Other topics related to π Tau v t e
Liu_Hui's_π_algorithm
Japanese mathematician and cartographer
(發微算法演段諺解) OCLC 22056085721 Sangaku, the custom of presenting mathematical problems, carved in wood tablets, to the public in shinto shrines Soroban, a Japanese
Takebe_Kenkō
Mengoli (1626–1686) – priest and mathematician who first posed the famous Basel Problem Giuseppe Mercalli (1850–1914) – priest, volcanologist, and director
List of Catholic clergy scientists
List_of_Catholic_clergy_scientists
Construction in functional analysis, useful to solve differential equations
}|f(x)|^{2}\,dx=\sum _{n=1}^{\infty }{\frac {1}{n^{2}}}.} Known as the Basel problem, this series converges to π 2 6 {\textstyle {\frac {\pi ^{2}}{6}}}
Decomposition of spectrum (functional analysis)
Decomposition_of_spectrum_(functional_analysis)
BASEL PROBLEM
BASEL PROBLEM
Boy/Male
Muslim/Islamic
Brave
Boy/Male
Arabic
Justice; Justify
Biblical
confusion; mixture,confusion,gate of God
Boy/Male
Arabic, Australian
Smiling
Boy/Male
Tamil
King, Basil the herb
Boy/Male
Hindu
King, Basil the herb
Surname or Lastname
English and French
English and French : from a medieval personal name, ultimately from Greek Basileios ‘royal’. The name was borne by a 4th-century bishop of Caesarea in Cappadocia, regarded as one of the four Fathers of the Eastern Church; he wrote important theological works and established a rule for religious orders of monks. Various other saints are also known under these and cognate names. The popularity of Vasili as a Russian personal name is largely due to the fact that this was the ecclesiastical name of St. Vladimir (956–1015), Prince of Kiev, who was chiefly responsible for the introduction of Christianity to Russia. As an American surname, this has also absorbed some Greek, Russian, and other derivatives of Greek Vasili.
Boy/Male
Arabic, Australian, Greek
Royal Kingly
Boy/Male
American, Arabic, Assamese, Bengali, British, Christian, English, French, German, Greek, Hindu, Indian, Irish, Lebanese, Muslim, Sindhi, Tamil
Kingly; Brave; Royal; The Great; King Like
Boy/Male
Australian, Jamaican
Manse of Clergyman
Boy/Male
Muslim/Islamic
Smiling
Boy/Male
Greek American English
Royal. Kingly. St Basil the Great was Bishop of Caesarea in the latter half of the 4th century....
Girl/Female
Biblical, British, English, French, Greek
Confusion; Mixture
Male
English
 English form of French Basile, BASIL means "king." Also sometimes given as an herb name.
Boy/Male
Afghan, African, Arabic, Australian, Chinese, Greek, Indian, Muslim
Brave
Boy/Male
Muslim
King, Basil the herb (1)
Boy/Male
Muslim
Smiling
Boy/Male
Muslim/Islamic
Brave
Boy/Male
Indian
Path guider
Girl/Female
Hindu, Indian
Royal; Kingly; The Great
BASEL PROBLEM
BASEL PROBLEM
Boy/Male
English Scandinavian
From the white farm.
Female
Babylonian
, goddess of creation.
Boy/Male
Latin
A Greek.
Surname or Lastname
English
English : habitational name from Codrington in Gloucestershire, named from the Old English personal name Cūþhere + -ing- denoting association with + tūn ‘settlement’.
Boy/Male
Hindu
Lord Vishnu
Girl/Female
Indian
Brisk, Swift
Girl/Female
Arabic, Muslim
Bright Minded
Boy/Male
Hindu
Chief, Worthy of admiration
Female
German
 Variant spelling of German Imma, IMA means "entire, whole." Compare with another form of Ima.
Girl/Female
Hindu
BASEL PROBLEM
BASEL PROBLEM
BASEL PROBLEM
BASEL PROBLEM
BASEL PROBLEM
n.
To put on a base or basis; to lay the foundation of; to found, as an argument or conclusion; -- used with on or upon.
a.
Morally low. Hence: Low-minded; unworthy; without dignity of sentiment; ignoble; mean; illiberal; menial; as, a base fellow; base motives; base occupations.
n.
The lowest member of a base when divided horizontally, or of a baseboard, pedestal, or the like.
n.
The bottom of anything, considered as its support, or that on which something rests for support; the foundation; as, the base of a statue.
a.
Deep or grave in sound; as, the base tone of a violin.
a.
Of little, or less than the usual, height; of low growth; as, base shrubs.
a.
Having the nerves radiating from the base; -- said of leaves.
a.
Alloyed with inferior metal; debased; as, base coin; base bullion.
a.
Not held by honorable service; as, a base estate, one held by services not honorable; held by villenage. Such a tenure is called base, or low, and the tenant, a base tenant.
adv.
In a base manner; with despicable meanness; dishonorably; shamefully.
n.
The name given to several aromatic herbs of the Mint family, but chiefly to the common or sweet basil (Ocymum basilicum), and the bush basil, or lesser basil (O. minimum), the leaves of which are used in cookery. The name is also given to several kinds of mountain mint (Pycnanthemum).
a.
Relating to, or forming, the base.
n.
A low, or deep, sound. (Mus.) (a) The lowest part; the deepest male voice. (b) One who sings, or the instrument which plays, base.
n.
Wearing, or protected by, bases.
n.
The basal plane of a crystal.
a.
Having a base, or having as a base; supported; as, broad-based.
n.
A rustic play; -- called also prisoner's base, prison base, or bars.
imp. & p. p.
of Base
n.
The number from which a mathematical table is constructed; as, the base of a system of logarithms.
n.
Same as Prison base.